Comparison of Stabilization of Piezoelectric Euler-Bernoulli Beam Models

Piezoelectric materials are used in many control and sensing applications; such as vibration control in small and large mechanical structures, shape control of beams, plates, and surfaces, and on-line measurement and compensation in aerospace and other high-precision areas. A strip of piezoelectric material is known as a piezoelectric beam. The behavior of piezoelectric beams depends on interactions between electrical, magnetic and mechanical effects and is rather complex. A number of different assumptions for the mechanical part of the beam and the electric-field can be used. For stabilization, and more generally, controller design, a finite-dimensional approximation must be used. In this work the linear Euler-Bernoulli model for the mechanical behavior is used with the quasi-static electric-field assumption, the dynamic electromagnetic-field assumption, and static case. The mathematical models are put in the port-Hamiltonian framework. The contribution of this work is the investigation of the influence of the type of approximation on stabilization and controlled system performance. The obtained models are compared to other (port-Hamiltonian) models under different modeling assumptions on the basis of different modeling assumptions and their influence on stabilization and controllability for the infinite- and finite-dimensional systems. The stability properties of the approximated linear port-Hamiltonian-model are investigated under the two assumptions for the electric-field and compared to the original infinite-dimensional model. The energy functions of the various models are also compared.